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Incoming links: Matrix addition
Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge 2003 Paper 1 5 c iv Solution Created 2026-10-18 Updated 2026-10-20
With The inner
dimensions as in part (i), the alternative definition isaddMat :: Num a => Matrix a -> Matrix a -> Matrix a
addMat a b
| dimensions a /= dimensions b = error "addMat: incompatible sizes"
| otherwise = zipWith (zipWith (+)) a bzipWith adds corresponding entries; the outer one applies that row operation to corresponding rows. This computes the same matrix addition, with the same rejection of unequal dimensions. Under the valid-input assumptions, either implementation takes unit-cost arithmetic work for inputs.Use the source's The outer list comprehension pairs corresponding rows; the inner one pairs corresponding entries. Checking both dimensions before using
type Matrix a = [[a]]. For its nonempty rectangular matrices, the number of rows and the first row's length determine the dimensions:dimensions :: Matrix a -> (Int,Int)
dimensions rows = (length rows, length (head rows))
addMat :: Num a => Matrix a -> Matrix a -> Matrix a
addMat a b
| dimensions a /= dimensions b = error "addMat: incompatible sizes"
| otherwise = [[x + y | (x,y) <- zip r s]
| (r,s) <- zip a b]zip prevents silent truncation. Thus the result has entries , the definition of matrix addition. The Num constraint supplies the addition operation.Generate successive matrix powers and take cumulative sums using matrix addition:For the stipulated square matrix,
seriesMat :: Num a => Matrix a -> [Matrix a]
seriesMat m = scanl1 addMat (iterate (multMat m) m)iterate (multMat m) m produces m, m*m, m*(m*m), and so on. Associativity of exact matrix multiplication identifies these as . The left scan scanl1 addMat then returnsAll dimensions remain compatible. The infinite outer list is available a finite prefix at a time under lazy evaluation; no convergence of the matrix geometric series is needed. With floating-point entries, the expression specifies this particular multiplication and addition order, since machine arithmetic need not obey the exact algebraic laws.The zero matrix has the required pattern, and matrix addition and scalar multiplication preserve every prescribed zero entry. This proves the vector subspace condition. A basis consists of the matrix unitsThey independently select every permitted entry of a tridiagonal matrix, so the dimension is for .